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AC equal to the two DE, DF, each to each, viz. AB equal to DE, Book L and AC to DF; but the angle BAC greater than the angle EDF.

the base BC is also greater than the base EF.

Of the two fides DE, DF, let DE be the fide which is not greater than the other, and at the point D in the straight line DE. make the angle EDG equal to the angle BAC; and make DG a. 23. I. equal b to AC or DF, and join EG, GF.

Because AB is equal to DE, and AC to DG, the two fides

BA, AC are equal to the two ED, DG, each to each, and the

b. 3. I.

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therefore the angle DFG is greater than EGF; and much more is the angle EFG greater than the angle EGF. and because the angle EFG of the triangle EFG is greater than its angle EGF, and that the greater fide is oppofite to the greater angle; the e. 19. . fide EG is therefore greater than the fide EF. but EG is equal" to BC; and therefore alfo BC is greater than EF. therefore if two triangles, &c. Q. E. D.

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F two triangles have two fides of the one equal to two fides of the other, each to each, but the base of the one greater than the base of the other; the angle alfo contained by the fides of that which has the greater base, shall be greater than the angle contained by the fides equal to them, of the other.

Let ABC, DEF be two triangles which have the two fides AB, AC equal to the two fides DE, DF, each to each, viz. AB equal to DE, and AC to DF; but the bafe CB greater than the base EF. the angle BAC is likewife greater than the angle EDF.

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Book I.

B. 4. I.

b. 24. I.

For if it be not greater, it must either be equal to it, or lefs. but the angle BAC is not equal to the angle EDF, because then the bafe BC would be

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gle BAC is not less than the angle EDF, and it was shewn that it is not equal to it; therefore the angle BAC is greater than the angle EDF, Wherefore if two triangles, &c. Q. E. D.

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F two triangles have two angles of one equal to twe angles of the other; each to each, and one fide equal to one fide, viz. either the fides adjacent to the equal angles, or the fides oppofite to equal angles in each; then fhall the other fides be equal, each to each, and also the third angle of the one to the third angle of the other.

Let ABC, DEF be two triangles which have the angles ABC, BCA equal to the angles DEF, EFD, viz. ABC to DEF, and BCA to EFD; alfo one fide equal to one fide; and firft, let thofe fides be equal which are adjacent to the angles that are equal in the two triangles, viz. BC to EF. the other fides fhall be equal, each to each, viz. AB to DE, and AC to DF; and the third angle BAC to the third angle EDF.

For if AB be not equal to DE, one of them must be the greater. Let AB be

A

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B

D

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and make BG equal to DE, and join GC, therefore because BG is

2

equal t› DE, and BC to EF, the two fides GB, BC are equal to Book I. the two DE, EF, each to each; and the angle GBC is equal to the angle DEF; therefore the base GC is equal to the base DF, and a. 4. 1. the tiangle GBC to the triangle DEF, and the other angles to the other angles, each to each, to which the equal fides are oppofite; therefore the angle GCB is equal to the angle DFE; but DFE is, by the hypothefis, equal to the angle BCA; wherefore also the angle BCG is equal to the angle BCA, the lefs to the greater, which is impoffible. therefore AB is not unequal to DE, that is, it is equal to it. and BC is equal to EF; therefore the two AB, BC are equal to the two DE, EF, each to each; and the angle ABC is equal to the angle DEF, the base therefore AC is equal a to the bafe DF, and the third angle BAC to the third angle EDF.

Next, let the fides

which are oppofite to
equal angles in each A
triangle be equal to
one arother, viz. AB
to DE; likewise in
this cafe, the other
fides (hall be equal,
AC to DF, and BC
to EF; and alfo the
third angle BAC to the third EDF.

B

D

HC E

F

For if BC be not equal to EF, let BC be the greater of them, and make BH equal to EF, and join AH. and because BH is equal to EF, and AB to DE; the two AB, BH are equal to the two DE, EF, each to each; and they contain equal angles; therefore the bafe AH is equal to the base DF, and the triangle ABH to the triangle DEF, and the other angles fhall be equal, each to each, to which the equal fides are oppofite. therefore the angle BHA is equal to the angle EFD. but EFD is equal to the angle BCA; therefore also the angle BHA is equal to the angle BCA, that is, the exterior angle BHA of the triangle AHC is equal to its interior and oppofite angle BCA; which is impoffible b. where- b. 16. 1. fore BC is not unequal to EF, that is, it is equal to it; and AB is equal to DE; therefore the two AB, BC are equal to the two DE, EF, each to each; and they contain equal angles; wherefore, the bafe AC is equal to the base DF, and the third angle BAC to the third angle EDF. therefore if two triangles, &c. Q2E, D.

Book I.

a. 16. 1.

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PROP. XXVII. THEOR.

a ftraight line falling upon two other straight lines makes the alternate angles equal to one another, these two straight lines fhall be parallel.

Let the ftraight line EF which falls upon the two straight lines AB, CD make the alternate angles AEF, EFD equal to one another; AB is parallel to CD.

For if it be not parallel, AB and CD being produced shall meet either towards BD or towards AC. let them be produced and meet towards BD in the point G; therefore GEF is a triangle, and its exterior angle AEF is greater than the interior and oppofite angle EFG; but it is alfo equal to it, which is impoffible, there

a

fore AB and CD being pro-A

E

B

duced do not meet towards
BD. in like manner it may be
demonftrated that they do not
meet towards AC. but those

ftraight lines which meet nei

D

b. 35. Def. ther way tho' produced ever fo far are parallel to one another. AB therefore is parallel to CD. wherefore if a straight line, &c. Q.E. D.

PROP. XXVIII. THEOR.

Fa ftraight line falling upon two other ftraight lines

I'

makes the exterior angle equal to the interior and oppofite upon the fame fide of the line; or makes the interior angles upon the fame fide together equal to two right angles; the two ftraight lines fhall be parallel to one another.

E

Let the straight line EF which falls upon the two straight lines
AB, CD make the exterior angle
EGB equal to the interior and op-
pofite angle GHD upon the fame
fide; or make the interior angles A

on the fame fide BGH, GHD to-
gether equal to two right angles. C
AB is parallel to CD.

Because the angle EGB is equal
to the angle GHD, and the angle

B

H

F

a

C

Book I.

a. 15. I. b. 27. I.

EGB equal to the angle AGH, the angle AGH is equal to the angle GHD; and they are the alternate angles; therefore AB is parallel to CD. again, because the angles BCH, GHD are equal to two right angles, and that AGH, BGH are also equal to two c. By Hyp. right angles; the angles AGH, BGH are equal to the angles BGH, d. 13. 1. GHD. take away the common angle BGH, therefore the remaining angle AGH is equal to the remaining angle GHD; and they are alternate angles; therefore AB is parallel to CD. wherefore if a straight line, &c. Q. E. D.

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PROP. XXIX. THEOR.

Notes on

F a ftraight line falls upon two parallel ftraight lines, See the it makes the alternate angles equal to one another; this Propoand the exterior angle equal to the interior and oppofite fition. upon the fame fide; and likewife the two interior angles upon the fame fide together equal to two right angles.

Let the straight line EF fall upon the parallel straight lines AB, CD. the alternate angles AGH, GHD are equal to one another; and the exterior angle EGB is equal to the interior and oppofite, upon the fame fide, GHD; and the two interior angles BGH, GHD upon the fame fide are together equal to two right angles.

For if AGH be not equal to GHD, one of them must be greater than the‐ other; let AGH be the greater. and becaufe the angle AGH is greater than the angle GHD, add to each of them

E

A

C

B

H

D

F

the angle BGH; therefore the angles AGH, BGH are greater than the angles BGH, GHD. but the angles AGH, BGH are equal a to a. 13. I. two right angles; therefore the angles BGH, GHD are less than two right angles. but those straight lines which with another straight line falling upon them make the interior angles on the fame fide less than two right angles, do meet * together if continually pro- * 12. Ax. duced; therefore the straight lines AB, CD if produced far enough fhall meet. but they never meet, fince they are parallel by the Hy pothesis. therefore the angle AGH is not unequal to the angle GHD, fition. that is, it is equal to it. but the angle AGH is equal to the angle b. 15. I. EGB; therefore likewife EGB is equal to GHD. add to each of

b

See the Notes on this Propo

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